Weightedness measures from inequality systems

A simple game is a cooperative game where some coalitions among players or voters became the (monotonic) set of winning coalitions, and the other ones form the set of losing coalitions. It is well-known that weighted voting games form a strict subclass of simple games, where each player has a voting weight so that a coalition wins if and only if the sum of weights of their members exceeds a given quota, otherwise it loses. This work studies how far away a simple game is for being representable as a weighted voting game, which allows for a more compact representation. There are several measures that determine the weightedness of a simple game, such as the dimension, the trade-robustness, the critical threshold value associated with the α -roughly weightedness property, etc. In this work we propose some new weightedness measures, all based on linear programming. In general terms, for a given simple game, a linear program is used to identify its weightedness: (i) the ϵ -roughly value ( μ ϵ ), (ii) the Z + -roughly value ( μ Z + ), (iii) the Δ-roughly value ( μ Δ ), and (iv) the outlier value ( Ψ M ). We show a close relation between the known critical threshold value of weightedness and the new measure μ Δ . Finally, we also present an exhaustive comparison of weightedness measures for simple games with up to six players. • We define new weightedness measures for simple games from inequality systems. • We introduce a new weightedness measure for simple games based on outlier coalitions. • This manuscript provides a comparison of the known α -roughly value with other new weightedness values. • We exhaustively compute some weightedness measures for all simple games up to 7 players.